Theorem mapss2

Description: Subset inheritance for set exponentiation. (Contributed by Glauco Siliprandi, 3-Mar-2021)

	Ref	Expression
Hypotheses	mapss2.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	mapss2.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
	mapss2.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑍 )
	mapss2.n	⊢ ( 𝜑 → 𝐶 ≠ ∅ )
Assertion	mapss2	⊢ ( 𝜑 → ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		mapss2.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2		mapss2.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
3		mapss2.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑍 )
4		mapss2.n	⊢ ( 𝜑 → 𝐶 ≠ ∅ )
5	2	adantr	⊢ ( ( 𝜑 ∧ 𝐴 ⊆ 𝐵 ) → 𝐵 ∈ 𝑊 )
6		simpr	⊢ ( ( 𝜑 ∧ 𝐴 ⊆ 𝐵 ) → 𝐴 ⊆ 𝐵 )
7		mapss	⊢ ( ( 𝐵 ∈ 𝑊 ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) )
8	5 6 7	syl2anc	⊢ ( ( 𝜑 ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) )
9	8	ex	⊢ ( 𝜑 → ( 𝐴 ⊆ 𝐵 → ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) )
10		n0	⊢ ( 𝐶 ≠ ∅ ↔ ∃ 𝑥 𝑥 ∈ 𝐶 )
11	4 10	sylib	⊢ ( 𝜑 → ∃ 𝑥 𝑥 ∈ 𝐶 )
12	11	adantr	⊢ ( ( 𝜑 ∧ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) → ∃ 𝑥 𝑥 ∈ 𝐶 )
13		eqidd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) → ( 𝑤 ∈ 𝐶 ↦ 𝑦 ) = ( 𝑤 ∈ 𝐶 ↦ 𝑦 ) )
14		eqidd	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) ∧ 𝑤 = 𝑥 ) → 𝑦 = 𝑦 )
15		simpr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) → 𝑥 ∈ 𝐶 )
16		vex	⊢ 𝑦 ∈ V
17	16	a1i	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) → 𝑦 ∈ V )
18	13 14 15 17	fvmptd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) → ( ( 𝑤 ∈ 𝐶 ↦ 𝑦 ) ‘ 𝑥 ) = 𝑦 )
19	18	eqcomd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) → 𝑦 = ( ( 𝑤 ∈ 𝐶 ↦ 𝑦 ) ‘ 𝑥 ) )
20	19	ad4ant13	⊢ ( ( ( ( 𝜑 ∧ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) ∧ 𝑥 ∈ 𝐶 ) ∧ 𝑦 ∈ 𝐴 ) → 𝑦 = ( ( 𝑤 ∈ 𝐶 ↦ 𝑦 ) ‘ 𝑥 ) )
21		simplr	⊢ ( ( ( 𝜑 ∧ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) )
22		simplr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑤 ∈ 𝐶 ) → 𝑦 ∈ 𝐴 )
23	22	fmpttd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ( 𝑤 ∈ 𝐶 ↦ 𝑦 ) : 𝐶 ⟶ 𝐴 )
24	1	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → 𝐴 ∈ 𝑉 )
25	3	adantr	⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → 𝐶 ∈ 𝑍 )
26	24 25	elmapd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ( ( 𝑤 ∈ 𝐶 ↦ 𝑦 ) ∈ ( 𝐴 ↑m 𝐶 ) ↔ ( 𝑤 ∈ 𝐶 ↦ 𝑦 ) : 𝐶 ⟶ 𝐴 ) )
27	23 26	mpbird	⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ( 𝑤 ∈ 𝐶 ↦ 𝑦 ) ∈ ( 𝐴 ↑m 𝐶 ) )
28	27	adantlr	⊢ ( ( ( 𝜑 ∧ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝑤 ∈ 𝐶 ↦ 𝑦 ) ∈ ( 𝐴 ↑m 𝐶 ) )
29	21 28	sseldd	⊢ ( ( ( 𝜑 ∧ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝑤 ∈ 𝐶 ↦ 𝑦 ) ∈ ( 𝐵 ↑m 𝐶 ) )
30		elmapi	⊢ ( ( 𝑤 ∈ 𝐶 ↦ 𝑦 ) ∈ ( 𝐵 ↑m 𝐶 ) → ( 𝑤 ∈ 𝐶 ↦ 𝑦 ) : 𝐶 ⟶ 𝐵 )
31	29 30	syl	⊢ ( ( ( 𝜑 ∧ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝑤 ∈ 𝐶 ↦ 𝑦 ) : 𝐶 ⟶ 𝐵 )
32	31	adantlr	⊢ ( ( ( ( 𝜑 ∧ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) ∧ 𝑥 ∈ 𝐶 ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝑤 ∈ 𝐶 ↦ 𝑦 ) : 𝐶 ⟶ 𝐵 )
33		simplr	⊢ ( ( ( ( 𝜑 ∧ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) ∧ 𝑥 ∈ 𝐶 ) ∧ 𝑦 ∈ 𝐴 ) → 𝑥 ∈ 𝐶 )
34	32 33	ffvelrnd	⊢ ( ( ( ( 𝜑 ∧ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) ∧ 𝑥 ∈ 𝐶 ) ∧ 𝑦 ∈ 𝐴 ) → ( ( 𝑤 ∈ 𝐶 ↦ 𝑦 ) ‘ 𝑥 ) ∈ 𝐵 )
35	20 34	eqeltrd	⊢ ( ( ( ( 𝜑 ∧ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) ∧ 𝑥 ∈ 𝐶 ) ∧ 𝑦 ∈ 𝐴 ) → 𝑦 ∈ 𝐵 )
36	35	ralrimiva	⊢ ( ( ( 𝜑 ∧ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) ∧ 𝑥 ∈ 𝐶 ) → ∀ 𝑦 ∈ 𝐴 𝑦 ∈ 𝐵 )
37		dfss3	⊢ ( 𝐴 ⊆ 𝐵 ↔ ∀ 𝑦 ∈ 𝐴 𝑦 ∈ 𝐵 )
38	36 37	sylibr	⊢ ( ( ( 𝜑 ∧ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) ∧ 𝑥 ∈ 𝐶 ) → 𝐴 ⊆ 𝐵 )
39	38	ex	⊢ ( ( 𝜑 ∧ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) → ( 𝑥 ∈ 𝐶 → 𝐴 ⊆ 𝐵 ) )
40	39	exlimdv	⊢ ( ( 𝜑 ∧ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) → ( ∃ 𝑥 𝑥 ∈ 𝐶 → 𝐴 ⊆ 𝐵 ) )
41	12 40	mpd	⊢ ( ( 𝜑 ∧ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) → 𝐴 ⊆ 𝐵 )
42	41	ex	⊢ ( 𝜑 → ( ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) → 𝐴 ⊆ 𝐵 ) )
43	9 42	impbid	⊢ ( 𝜑 → ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ↑m 𝐶 ) ⊆ ( 𝐵 ↑m 𝐶 ) ) )